Basis pursuit for geophysical inversion

Brad Artman and Mauricio Sacchi

brad@sep.stanford.edu, sacchi@phys.ualberta.ca

ABSTRACT

Accepting an inversion principle, it is possible to design an algorithm to meet any requirements or constraints. Given the context of representing a signal with an arbitrary overcomplete dictionary of waveforms within the signal, one can design an inversion algorithm that will focus energy into a small number of model space coefficients. With this principle in mind, an analogy to linear programming is developed that results in an algorithm with: the properties of an l¹ norm, a small and stable parameter space, definable convergence, and impressive denoising capabilities. Linear programming methods solve a nonlinear problem in an interior/barrier loop method similar to iteratively reweighted least squares (IRLS) algorithms, and are much slower than a least squares solution obtained with a conjugate gradient method. Velocity scanning with the hyperbolic radon transform is implemented as a test case for the methodology.